16,138 research outputs found

    A proof of the stability of extremal graphs, Simonovits' stability from Szemer\'edi's regularity

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    The following sharpening of Tur\'an's theorem is proved. Let Tn,pT_{n,p} denote the complete pp--partite graph of order nn having the maximum number of edges. If GG is an nn-vertex Kp+1K_{p+1}-free graph with e(Tn,p)−te(T_{n,p})-t edges then there exists an (at most) pp-chromatic subgraph H0H_0 such that e(H0)≄e(G)−te(H_0)\geq e(G)-t. Using this result we present a concise, contemporary proof (i.e., one applying Szemer\'edi's regularity lemma) for the classical stability result of Simonovits.Comment: 4 pages plus reference

    Symmetric Graph Convolutional Autoencoder for Unsupervised Graph Representation Learning

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    We propose a symmetric graph convolutional autoencoder which produces a low-dimensional latent representation from a graph. In contrast to the existing graph autoencoders with asymmetric decoder parts, the proposed autoencoder has a newly designed decoder which builds a completely symmetric autoencoder form. For the reconstruction of node features, the decoder is designed based on Laplacian sharpening as the counterpart of Laplacian smoothing of the encoder, which allows utilizing the graph structure in the whole processes of the proposed autoencoder architecture. In order to prevent the numerical instability of the network caused by the Laplacian sharpening introduction, we further propose a new numerically stable form of the Laplacian sharpening by incorporating the signed graphs. In addition, a new cost function which finds a latent representation and a latent affinity matrix simultaneously is devised to boost the performance of image clustering tasks. The experimental results on clustering, link prediction and visualization tasks strongly support that the proposed model is stable and outperforms various state-of-the-art algorithms.Comment: 10 pages, 3 figures, ICCV 2019 accepte

    Nucleation and growth in two dimensions

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    We consider a dynamical process on a graph GG, in which vertices are infected (randomly) at a rate which depends on the number of their neighbours that are already infected. This model includes bootstrap percolation and first-passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph Z2\mathbb{Z}^2, significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.Comment: 35 pages, Section 6 update

    Graphs based methods for simultaneous smoothing and sharpening

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    [EN] We present two new methods for simultaneous smoothing and sharpening of color images: the GMS(3) (Graph Method for Simultaneous Smoothing and Sharpening) and the NGMS(3)(Normalized Graph-Method for Simultaneous Smoothing and Sharpening). They are based on analyzing the structure of local graphs computed at every pixel using their respective neighbors. On the one hand, we define a kernel-based filter for smoothing each pixel with the pixels associated to nodes in its same connected component. On the other hand, we modify each pixel by increasing their differences with respect to the pixels in the other connected components of those local graphs. Our approach is shown to be competitive with respect to other state-of-the-art methods that simultaneously manage both processes. We provide two methods that carry out the process of smoothing and sharpening simultaneously. The methods are based on the analysis of the structure of a local graph defined from the differences in the RGB space among the pixels in a 3 x 3 window. The parameters of the method are adjusted using both observers opinion and the well-known reference image quality assessment BRISQUE (Blind/Referenceless images spatial quality Evaluator) score.C. Jordan acknowledges the support of grant TEC2016-79884-C2-2-R. S. Morillas acknowledges the support of grant MTM2015-64373-P (MINECO/FEDER, UE).PĂ©rez-Benito, C.; Jordan-Lluch, C.; Conejero, JA.; Morillas, S. (2020). Graphs based methods for simultaneous smoothing and sharpening. MethodsX. 7:1-5. https://doi.org/10.1016/j.mex.2020.100819S15

    Techniques de bundling : un cas d'étude pour l'exploration de grandes quantités d'informations

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    We present a fast and simple method to compute bundled layouts of general graphs, dynamic graphs and temporal paths. For this, we first transform a given graph drawing into a density map using kernel density estimation. Next, we apply an image sharpening technique which progressively merges local height maxima by moving the convolved graph edges into the height gradient flow. We show how these techniques can produce simplified visualizations of static, streaming and sequence graphs. We illustrate our techniques with datasets from aircraft monitoring, software engineering, and eye-tracking of static and dynamic scenes
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